3,522 research outputs found
Worst case and probabilistic analysis of the 2-Opt algorithm for the TSP
2-Opt is probably the most basic local search heuristic for the TSP. This heuristic achieves amazingly good results on “real world” Euclidean instances both with respect to running time and approximation ratio. There are numerous experimental studies on the performance of 2-Opt. However, the theoretical knowledge about this heuristic is still very limited. Not even its worst case running time on 2-dimensional Euclidean instances was known so far. We clarify this issue by presenting, for every p∈N , a family of L p instances on which 2-Opt can take an exponential number of steps.
Previous probabilistic analyses were restricted to instances in which n points are placed uniformly at random in the unit square [0,1]2, where it was shown that the expected number of steps is bounded by O~(n10) for Euclidean instances. We consider a more advanced model of probabilistic instances in which the points can be placed independently according to general distributions on [0,1] d , for an arbitrary d≥2. In particular, we allow different distributions for different points. We study the expected number of local improvements in terms of the number n of points and the maximal density ϕ of the probability distributions. We show an upper bound on the expected length of any 2-Opt improvement path of O~(n4+1/3⋅ϕ8/3) . When starting with an initial tour computed by an insertion heuristic, the upper bound on the expected number of steps improves even to O~(n4+1/3−1/d⋅ϕ8/3) . If the distances are measured according to the Manhattan metric, then the expected number of steps is bounded by O~(n4−1/d⋅ϕ) . In addition, we prove an upper bound of O(ϕ√d) on the expected approximation factor with respect to all L p metrics.
Let us remark that our probabilistic analysis covers as special cases the uniform input model with ϕ=1 and a smoothed analysis with Gaussian perturbations of standard deviation σ with ϕ∼1/σ d
Optimal Online Edge Coloring of Planar Graphs with Advice
Using the framework of advice complexity, we study the amount of knowledge
about the future that an online algorithm needs to color the edges of a graph
optimally, i.e., using as few colors as possible. For graphs of maximum degree
, it follows from Vizing's Theorem that bits of
advice suffice to achieve optimality, where is the number of edges. We show
that for graphs of bounded degeneracy (a class of graphs including e.g. trees
and planar graphs), only bits of advice are needed to compute an optimal
solution online, independently of how large is. On the other hand, we
show that bits of advice are necessary just to achieve a
competitive ratio better than that of the best deterministic online algorithm
without advice. Furthermore, we consider algorithms which use a fixed number of
advice bits per edge (our algorithm for graphs of bounded degeneracy belongs to
this class of algorithms). We show that for bipartite graphs, any such
algorithm must use at least bits of advice to achieve
optimality.Comment: CIAC 201
Fast Algorithm for Partial Covers in Words
A factor of a word is a cover of if every position in lies
within some occurrence of in . A word covered by thus
generalizes the idea of a repetition, that is, a word composed of exact
concatenations of . In this article we introduce a new notion of
-partial cover, which can be viewed as a relaxed variant of cover, that
is, a factor covering at least positions in . We develop a data
structure of size (where ) that can be constructed in time which we apply to compute all shortest -partial covers for a
given . We also employ it for an -time algorithm computing
a shortest -partial cover for each
Online Bin Covering: Expectations vs. Guarantees
Bin covering is a dual version of classic bin packing. Thus, the goal is to
cover as many bins as possible, where covering a bin means packing items of
total size at least one in the bin.
For online bin covering, competitive analysis fails to distinguish between
most algorithms of interest; all "reasonable" algorithms have a competitive
ratio of 1/2. Thus, in order to get a better understanding of the combinatorial
difficulties in solving this problem, we turn to other performance measures,
namely relative worst order, random order, and max/max analysis, as well as
analyzing input with restricted or uniformly distributed item sizes. In this
way, our study also supplements the ongoing systematic studies of the relative
strengths of various performance measures.
Two classic algorithms for online bin packing that have natural dual versions
are Harmonic and Next-Fit. Even though the algorithms are quite different in
nature, the dual versions are not separated by competitive analysis. We make
the case that when guarantees are needed, even under restricted input
sequences, dual Harmonic is preferable. In addition, we establish quite robust
theoretical results showing that if items come from a uniform distribution or
even if just the ordering of items is uniformly random, then dual Next-Fit is
the right choice.Comment: IMADA-preprint-c
Probabilistic alternatives for competitive analysis
In the last 20 years competitive analysis has become the main tool for analyzing the quality of online algorithms. Despite of this, competitive analysis has also been criticized: it sometimes cannot discriminate between algorithms that exhibit significantly different empirical behavior or it even favors an algorithm that is worse from an empirical point of view. Therefore, there have been several approaches to circumvent these drawbacks. In this survey, we discuss probabilistic alternatives for competitive analysis.operations research and management science;
Dynamic Balanced Graph Partitioning
This paper initiates the study of the classic balanced graph partitioning
problem from an online perspective: Given an arbitrary sequence of pairwise
communication requests between nodes, with patterns that may change over
time, the objective is to service these requests efficiently by partitioning
the nodes into clusters, each of size , such that frequently
communicating nodes are located in the same cluster. The partitioning can be
updated dynamically by migrating nodes between clusters. The goal is to devise
online algorithms which jointly minimize the amount of inter-cluster
communication and migration cost.
The problem features interesting connections to other well-known online
problems. For example, scenarios with generalize online paging, and
scenarios with constitute a novel online variant of maximum matching. We
present several lower bounds and algorithms for settings both with and without
cluster-size augmentation. In particular, we prove that any deterministic
online algorithm has a competitive ratio of at least , even with significant
augmentation. Our main algorithmic contributions are an -competitive deterministic algorithm for the general setting with
constant augmentation, and a constant competitive algorithm for the maximum
matching variant
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